Algorithms and Programming: Problems and Solutions (2nd by Alexander Shen

By Alexander Shen

"Algorithms and Programming" is basically meant for a primary 12 months undergraduate direction in programming. established in a problem-solution layout, the textual content motivates the coed to imagine in the course of the programming approach, therefore constructing a company figuring out of the underlying conception. even if a reasonable familiarity with programming is believed, the publication is well used by scholars new to laptop technology. The extra complex chapters make the publication precious for a graduate path within the research of algorithms and/or compiler construction.

New to the second one variation are extra chapters on suffix bushes, video games and techniques, and Huffman coding in addition to an appendix illustrating the convenience of conversion from Pascal to C. the fabric covers such issues as combinatorics, sorting, looking out, queues, grammar and parsing, chosen recognized algorithms, and lots more and plenty extra.

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Additional info for Algorithms and Programming: Problems and Solutions (2nd Edition) (Springer Undergraduate Texts in Mathematics and Technology)

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X[k], y[1] < . . < y[l]). Find the number of common elements in both arrays; that is, the number of integers t such that t = x[i] = y[j] for some i and j. ) Solution. ,y[l]} while (k1 <> k) and (l1 <> l) do begin if x[k1+1] < y[l1+1] then begin k1 := k1 + 1; end else if x[k1+1] > y[l1+1] then begin l1 := l1 + 1; end else begin {x[k1+1] = y[l1+1]} k1 := k1 + 1; l1 := l1 + 1; n := n + 1; end; end; {k1 = k or l1 = l; therefore, one of the sets mentioned in the invariant relation is empty and n is the number in question} Remark.

All sequences of length n that contain all the numbers 1 . . n). Solution. x[n]. Permutations are printed in lexicographic order. ni. 2 1i. How do we find the next permutation in the lexicographic order? When is it possible to increase k-th term in a permutation without changing all preceding terms? , terms with larger indices). Therefore, to find the next permutation we should find the maximum k where increase is possible; that is, a k such that x[k] < x[k+1] > . . > x[n] Next we increase x[k] but keep the increase as small as possible.

Write a program that finds P(n) for a given n. Solution. One can prove the following (nontrivial) formula for P(n): P(n) = P(n 1)+ P(n 2) P(n 5) P(n 7)+ P(n 12)+ P(n 15) + · · · (terms are grouped in pairs, the signs before the pairs alternate, arguments in q-th pair are n (3q 2 q)/2 and n (3q 2 + q)/2). We assume P(k) = 0 for k 6 0, so the sum is finite. Even if we did not know this formula, there is a way to compute P(n) that is much more efficient than counting all the partitions one-by-one.

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